1. Membrane Trafficking: a) Membrane wave: In the central dogma of endocytosis, ligand-receptor bindings stimulate upstream signals at PM, which triggers multiple cascades of early responses that finally converge to actin cytoskeleton nucleation and endocytic apparatus assembly. However, exactly what these early responses at the PM look like remains unknown. Our experiments demonstrate that antigen stimulation of mast cells (RBL-2H3) results in cycles of recruitment and shedding of actin regulatory proteins and endocytic proteins (FBP17, endophilin, FCHo, and clathrin) in the form of traveling waves at PM. We establish a theoretical model that underscores the well-known membrane curvature-sensitivities of endocytic proteins and their biochemical pathways in response to upstream stimulations. Our model shows that the traveling waves of the endocytic proteins mirror the local membrane shape changes. While a weaker upstream stimulation leads to traveling wave, a stronger one will result in a stable membrane tubule, upon which endocytosis begins. Our experiments verify such model predictions. These findings reveal a checkpoint control in the initiation of endocytosis. The intrinsic couplings among curvature-sensitive endocytic proteins dictate a threshold upstream stimulation, only above which, endocytosis can proceed. The traveling wave of endocytic proteins at PM, on the other hand, reflects weaker stimulations, and the fact that cell utilizes such internal control to cope with external cue to avoid unnecessary cellular processes. We hypothesize that such a mechanism can be of great implication in many other endomembrane trafficking processes, e.g., phagocytosis. We are currently writing this paper. b) Phagocytosis: During phagocytosis, cell extends its filapodia around the target, forming cup-shaped membrane invagination that subsequently seals at its distal margins to form intracellular, membrane-bounded organelle. In contrast to endocytosis, phagocytosis is strongly modulated by the size, the shape, and even the stiffness of the target. Such dependence has been utilized both by bacteria and host cells for entry and its prevention, respectively. In addition, lipids (e.g., PI(4,5)P2, DAG, and PI(3,4,5)P3) and actin regulatory proteins (e.g., Cdc42, WASP, and PKCs) form distinct gradients along the surface of phagocytic cup. And such spatial patterns evolve as the phagocytic cup seals its distal end. Despite intensive studies, underlying physical mechanism of phagocytosis is largely unknown. We establish a coherent theoretical model of phagocytosis. The key model ingredient is the coupling between the membrane curvature-sensitivity of phosphoinositide lipid enzymatic activities and membrane protrusion mediated by actin polymerizations. This model not only quantitatively accounts for the aforementioned peculiar nature of phagocytosis, but also unravels their commonality. Thus, our model suggests an evolutionary path along which endocytosis, phagocytosis, and other endomembrane trafficking events diverge. This paper is submitted to PNAS. 2. Cell Division: a) Spatial-temporal regulation of spindle assembly checkpoint: We establish a theoretical model that describes the mitotic spindle structure (chromosomes, mitotic spindles, and the spindle poles) as a coherent transport system. Depending on whether the chromosome is properly attached, SAC and cyclin B circulate within transport system by dynein, and exchange with cytoplasm in accordance to their well-known biochemical regulations. The basis of such coherent transport is the compartmentalization underscored by direct and indirect protein affinities with these mitotic structures. Our model results show that such transport system recapitulates the observed spatial-temporal pattern of SAC and cyclin B. Moreover, the transport system dictates a stepwise SP accumulation of these proteins, which tightly couples with each proper chromosome attachment event. Furthermore, the jump in SP accumulation increases as the number of unattached chromosomes decreases: the last chromosome attachment results in a jump in SP accumulation > two folds of that from the second-to-last attachment. Finally, our model predicts that cyclin B degradation starts at SP, which is consistent with experiment observation. Such degradation always couples with the last chromosome attachment event, and remains robust against typical fluctuations due to the large boost in SP accumulation. Our model thus provides a robust mechanism for silencing SAC activity in accordance to the last chromosome attachment. This paper is currently being drafted. 3. Cell Motility: a) Mechanochemistry of focal adhesion formation: Focal adhesions are essential to mediate cell extracellular matrix (ECM) adhesion and force transmission during cell motilities, which involve the crosstalk between physical signals such as contractile forces or membrane dynamics, and chemical signaling events such as focal adhesion kinase related regulation pathways. However, the underline mechanism of the biophysical regulations of force transmission among actin cytoskeleton, cell membrane, focal complex and ECM remains poorly understood. We collaborated with Dr. Clare Waterman, and constructed a mathematical model to understand the behavior of focal adhesion complex under different experimental conditions. By integrating the cell membrane dynamics, actin network fluid dynamics, and the mechanochemistry of focal complex, the model reveals itself the capability to capture the essential characteristics of focal adhesions in cell motility. In particular, the model explains the oscillation of traction force both in terms of position and magnitude within focal adhesions at different ECM stiffness. The model thus provides a comprehensive vision of the focal adhesion dynamics. This paper is currently in preparation for publication.